# American Institute of Mathematical Sciences

March  2019, 18(2): 977-998. doi: 10.3934/cpaa.2019048

## A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results

 1 Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy 2 Istituto di Matematica Applicata e Tecnologie Informatiche "Enrico Magenes", CNR, Via Ferrata 1, 27100 Pavia, Italy 3 Dipartimento di Ingegneria e Architettura, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy 4 Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy 5 Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Via dei Taurini 19, 00185 Roma, Italy

* Corresponding author

Received  February 2018 Revised  June 2018 Published  October 2018

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.

Citation: Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, Roberto Natalini. A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results. Communications on Pure & Applied Analysis, 2019, 18 (2) : 977-998. doi: 10.3934/cpaa.2019048
##### References:

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##### References:
Evolution of $c$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
Evolution of $s$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
Evolution of $r$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
Evolution of $c$ along a horizontal line within the solid located at a) $x_2 = 0.25$ and b) $x_2 = 0.75$ assuming parabolic relationship for $\nu (r)$
Evolution of $s$ along a horizontal line within the solid located at a) $x_2 = 0.25$ and b) $x_2 = 0.75$ assuming parabolic relationship for $\nu (r)$
Concentration of $SO_2$ within the solid at different time step a) $n = 5$ and b) $n = 15$ assuming parabolic relationship for $\nu (r)$
Evolution of $r$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
Evolution of $c$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
Evolution of $s$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
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